Question

The number of solutions of the equation $\cos \left(\pi \sqrt{x-4}\right)\cdot \cos \pi \sqrt{x}=1$ is

• A. $1$
• B. $2$
• C. More than two
• D. None of these

Answer 1

Answer: (A)

$1$

$\cos \left(\pi \sqrt{x-4}\right)\cdot \cos \pi \sqrt{x}=1.....(\ast )$
is possible only when
$\cos \left(\pi \sqrt{x-4}\right)=1$ and $\cos \left(\pi \sqrt{}\right)=1.....\left(i\right)$
Since $\sqrt{x-4}\ge 0$ and $\sqrt{x}\ge 0$
$\therefore x\ge 4$
From (i), $\pi \left(\sqrt{x-4}\right)=0$ and $\pi \sqrt{x}=0$
$\Rightarrow x=4$ and $x=0$
$\therefore x=4$ is only solution is (i)
$\therefore$ Choice (a) is correct answer.